$U(\frak h)$-free modules over the Lie algebras of differential operators

TL;DR

This paper classifies certain non-weight modules over Lie algebras of differential operators, focusing on modules that are free over a Cartan subalgebra and analyzing their tensor products and irreducibility.

Contribution

It provides a complete characterization of $U(rak h)$-free modules over Lie algebras of differential operators and their tensor product structures.

Findings

Identified modules free over $U(rak h)$ with rank 1 over differential operators on the circle.

Established conditions for irreducibility of tensor products of quasi-finite highest weight modules and $U(rak h)$-free modules.

Extended results to general Lie algebras of differential operators.

Abstract

In this paper, we consider some non-weight modules over the Lie algebra of Weyl type. First, we determine the modules whose restriction to $U(\frak h)$ are free of rank $1$ over the Lie algebra of differential operators on the circle. Then we determine the necessary and sufficient conditions for the tensor products of quasi-finite highest weight modules and $U(\frak h)$-free modules to be irreducible, and obtain that any two such tensor products are isomorphic if and only if the corresponding highest weight modules and $U(\frak h)$-free modules are isomorphic. Finally, we extend such results to the Lie algebras of differential operators in the general case.